Given a partition l and a composition b, the stretched Kostka coefficient K_{l, b}(n) is the map sending each positive integer n to the Kostka coefficient indexed by nl and nb. Kirillov and Reshetikhin (1986) have shown that stretched Kostka coefficients are polynomial functions of n. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients (2004), and they have given a conjectural expression for their degrees (2006). We prove the values conjectured by King, Tollu, and Toumazet for the degrees of stretched Kostka coefficients. Our proof depends upon the polyhedral geometry of Gelfand--Tsetlin polytopes and uses tilings of GT-patterns, a combinatorial structure introduced in De Loera & McAllister (2004).

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial time algorithm, based on counting the lattice points in polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based on the ellipsoid algorithm, to decide when the coefficients are nonzero in polynomial time for arbitrary rank. Our experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank. Using an implementation of this algorithm, we provide experimental evidence for conjectured generalizations of the saturation property of Littlewood--Richardson coefficients. One of these conjectures seems to be valid for types B_n, C_n, and D_n.

If $P\subset \R^d$ is a rational polytope, then $i_P(n):=#(nP\cap \Z^d)$ is a quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period of $i_P(n)$ must divide $\LL(P)= \min \{n \in \Z_{> 0} \colon nP \text{is an integral polytope}\}$. Few examples are known where the period is not exactly $\LL(P)$. We show that for any $\LL$, there is a 2-dimensional triangle $P$ such that $\LL(P)=\LL$ but such that the period of $i_P(n)$ is 1, that is, $i_P(n)$ is a polynomial in $n$. We also characterize all polygons $P$ such that $i_P(n)$ is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\mathfrak{gl}_n \C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.